Reflectance

Last updated December 07, 2023

The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

Mathematical definitions

Hemispherical reflectance

The hemispherical reflectance of a surface, denoted $R$ , is defined as^[1]

R={\frac {\Phi _{\mathrm {e} }^{\mathrm {r} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}},

where $Φ e r$ is the radiant flux reflected by that surface and $Φ e i$ is the radiant flux received by that surface.

Spectral hemispherical reflectance

The spectral hemispherical reflectance in frequency and spectral hemispherical reflectance in wavelength of a surface, denoted $R ν$ and $R λ$ respectively, are defined as^[1]

R_{\nu }={\frac {\Phi _{\mathrm {e} ,\nu }^{\mathrm {r} }}{\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}},

R_{\lambda }={\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {r} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}},

where

$Φ e, ν r$ is the spectral radiant flux in frequency reflected by that surface;
$Φ e, ν i$ is the spectral radiant flux in frequency received by that surface;
$Φ e, λ r$ is the spectral radiant flux in wavelength reflected by that surface;
$Φ e, λ i$ is the spectral radiant flux in wavelength received by that surface.

Directional reflectance

The directional reflectance of a surface, denoted R_Ω, is defined as^[1]

R_{\Omega }={\frac {L_{\mathrm {e} ,\Omega }^{\mathrm {r} }}{L_{\mathrm {e} ,\Omega }^{\mathrm {i} }}},

where

$L e,Ω r$ is the radiance reflected by that surface;
$L e,Ω i$ is the radiance received by that surface.

This depends on both the reflected direction and the incoming direction. In other words, it has a value for every combination of incoming and outgoing directions. It is related to the bidirectional reflectance distribution function and its upper limit is 1. Another measure of reflectance, depending only on the outgoing direction, is I/F, where I is the radiance reflected in a given direction and F is the incoming radiance averaged over all directions, in other words, the total flux of radiation hitting the surface per unit area, divided by π.^[2] This can be greater than 1 for a glossy surface illuminated by a source such as the sun, with the reflectance measured in the direction of maximum radiance (see also Seeliger effect).

Spectral directional reflectance

The spectral directional reflectance in frequency and spectral directional reflectance in wavelength of a surface, denoted $R Ω, ν$ and $R Ω, λ$ respectively, are defined as^[1]

R_{\Omega ,\nu }={\frac {L_{\mathrm {e} ,\Omega ,\nu }^{\mathrm {r} }}{L_{\mathrm {e} ,\Omega ,\nu }^{\mathrm {i} }}},

R_{\Omega ,\lambda }={\frac {L_{\mathrm {e} ,\Omega ,\lambda }^{\mathrm {r} }}{L_{\mathrm {e} ,\Omega ,\lambda }^{\mathrm {i} }}},

where

$L e,Ω, ν r$ is the spectral radiance in frequency reflected by that surface;
$L e,Ω, ν i$ is the spectral radiance received by that surface;
$L e,Ω, λ r$ is the spectral radiance in wavelength reflected by that surface;
$L e,Ω, λ i$ is the spectral radiance in wavelength received by that surface.

Again, one can also define a value of $I / F$ (see above) for a given wavelength.^[3]

Reflectivity

For homogeneous and semi-infinite (see halfspace) materials, reflectivity is the same as reflectance. Reflectivity is the square of the magnitude of the Fresnel reflection coefficient,^[4] which is the ratio of the reflected to incident electric field;^[5] as such the reflection coefficient can be expressed as a complex number as determined by the Fresnel equations for a single layer, whereas the reflectance is always a positive real number.

For layered and finite media, according to the CIE,^{[ citation needed ]} reflectivity is distinguished from reflectance by the fact that reflectivity is a value that applies to thick reflecting objects.^[6] When reflection occurs from thin layers of material, internal reflection effects can cause the reflectance to vary with surface thickness. Reflectivity is the limit value of reflectance as the sample becomes thick; it is the intrinsic reflectance of the surface, hence irrespective of other parameters such as the reflectance of the rear surface. Another way to interpret this is that the reflectance is the fraction of electromagnetic power reflected from a specific sample, while reflectivity is a property of the material itself, which would be measured on a perfect machine if the material filled half of all space.^[7]

Surface type

Given that reflectance is a directional property, most surfaces can be divided into those that give specular reflection and those that give diffuse reflection.

For specular surfaces, such as glass or polished metal, reflectance is nearly zero at all angles except at the appropriate reflected angle; that is the same angle with respect to the surface normal in the plane of incidence, but on the opposing side. When the radiation is incident normal to the surface, it is reflected back into the same direction.

For diffuse surfaces, such as matte white paint, reflectance is uniform; radiation is reflected in all angles equally or near-equally. Such surfaces are said to be Lambertian.

Most practical objects exhibit a combination of diffuse and specular reflective properties.

Water reflectance

Reflection occurs when light moves from a medium with one index of refraction into a second medium with a different index of refraction.

Specular reflection from a body of water is calculated by the Fresnel equations.^[8] Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection.

A real water surface may be wavy. Reflectance, which assumes a flat surface as given by the Fresnel equations, can be adjusted to account for waviness.

Grating efficiency

The generalization of reflectance to a diffraction grating, which disperses light by wavelength, is called diffraction efficiency .

Other radiometric coefficients

Radiometry coefficients
v
t
e
Quantity		SI units	Notes
Name	Sym.	SI units	Notes
Hemispherical emissivity	$ε$	—	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	$ε ν$ $ε λ$	—	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	$ε Ω$	—	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	$ε Ω, ν$ $ε Ω, λ$	—	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	$A$	—	Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	$A ν$ $A λ$	—	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	$A Ω$	—	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	$A Ω, ν$ $A Ω, λ$	—	Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance	$R$	—	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	$R ν$ $R λ$	—	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	$R Ω$	—	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	$R Ω, ν$ $R Ω, λ$	—	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	$T$	—	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	$T ν$ $T λ$	—	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	$T Ω$	—	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	$T Ω,ν$ $T Ω, λ$	—	Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient	$μ$	m⁻¹	Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	$μ ν$ $μ λ$	m⁻¹	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	$μ Ω$	m⁻¹	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	$μ Ω, ν$ $μ Ω, λ$	m⁻¹	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

Related Research Articles

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In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature $T$ , when there is no net flow of matter or energy between the body and its environment.

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Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample ". Alternatively, for samples which scatter light, absorbance may be defined as "the negative logarithm of one minus absorptance, as measured on a uniform sample". The term is used in many technical areas to quantify the results of an experimental measurement. While the term has its origin in quantifying the absorption of light, it is often entangled with quantification of light which is “lost” to a detector system through other mechanisms. What these uses of the term tend to have in common is that they refer to a logarithm of the ratio of a quantity of light incident on a sample or material to that which is detected after the light has interacted with the sample.

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In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, and to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre. It is a directional quantity: the radiance of a surface depends on the direction from which it is being observed.

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<span class="mw-page-title-main">Isotropic radiator</span>

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In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), one joule per second, while that of spectral flux in frequency is the watt per hertz and that of spectral flux in wavelength is the watt per metre —commonly the watt per nanometre.

In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre, while that of spectral exitance in frequency is the watt per square metre per hertz (W·m⁻²·Hz⁻¹) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m⁻³)—commonly the watt per square metre per nanometre. The CGS unit erg per square centimeter per second is often used in astronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

In radiometry, radiosity is the radiant flux leaving a surface per unit area, and spectral radiosity is the radiosity of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiosity is the watt per square metre, while that of spectral radiosity in frequency is the watt per square metre per hertz (W·m⁻²·Hz⁻¹) and that of spectral radiosity in wavelength is the watt per square metre per metre (W·m⁻³)—commonly the watt per square metre per nanometre. The CGS unit erg per square centimeter per second is often used in astronomy. Radiosity is often called intensity in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss. The SI unit of attenuation coefficient is the reciprocal metre (m⁻¹). Extinction coefficient is another term for this quantity, often used in meteorology and climatology. Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward e-folding distance of the original intensity as the energy of the intensity passes through a unit thickness of material, so that an attenuation coefficient of 1 m⁻¹ means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m⁻¹, it will be reduced twice by e, or e². Other measures may use a different factor than e, such as the decadic attenuation coefficient below. The broad-beam attenuation coefficient counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding. The mass attenuation coefficient is the attenuation coefficient normalized by the density of the material.

In the study of heat transfer, absorptance of the surface of a material is its effectiveness in absorbing radiant energy. It is the ratio of the absorbed to the incident radiant power.

In radiometry, radiant exposure or fluence is the radiant energy received by a surface per unit area, or equivalently the irradiance of a surface, integrated over time of irradiation, and spectral exposure is the radiant exposure per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant exposure is the joule per square metre, while that of spectral exposure in frequency is the joule per square metre per hertz and that of spectral exposure in wavelength is the joule per square metre per metre —commonly the joule per square metre per nanometre.

References

1 2 3 4 "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
↑ Cuzzi, Jeffrey; Chambers, Lindsey; Hendrix, Amanda (Oct 21, 2016). "Rough Surfaces: is the dark stuff just shadow?". Icarus . 289: 281–294. doi:10.1016/j.icarus.2016.10.018. PMC 6839776 . PMID 31708591.
↑ See for example P.G.J Irwin; et al. (Jan 12, 2022). "Hazy Blue Worlds: A Holistic Aerosol Model for Uranus and Neptune, Including Dark Spots". Journal of Geophysical Research: Planets. 127 (6): e2022JE007189. arXiv: 2201.04516 . Bibcode: 2022JGRE..12707189I . doi: 10.1029/2022JE007189 . hdl: 1983/65ee78f0-1d28-4017-bbd9-1b49b24700d7 . PMC 9286428 . PMID 35865671. S2CID 245877540 .
↑ E. Hecht (2001). Optics (4th ed.). Pearson Education. ISBN 0-8053-8566-5.
↑ IUPAC , Compendium of Chemical Terminology , 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) " Reflectance ". doi : 10.1351/goldbook.R05235
↑ "CIE International Lighting Vocabulary". Archived from the original on 2016-06-16. Retrieved 2010-12-04.
↑ Palmer and Grant, The Art of Radiometry
↑ Ottaviani, M. and Stamnes, K. and Koskulics, J. and Eide, H. and Long, S.R. and Su, W. and Wiscombe, W., 2008: 'Light Reflection from Water Waves: Suitable Setup for a Polarimetric Investigation under Controlled Laboratory Conditions . Journal of Atmospheric and Oceanic Technology, 25 (5), 715--728.

External links

Reflectivity of metals Archived 2016-03-04 at the Wayback Machine .
Reflectance Data.

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[ISO_9288-1989-1] 1 2 3 4 "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.

[2] Cuzzi, Jeffrey; Chambers, Lindsey; Hendrix, Amanda (Oct 21, 2016). "Rough Surfaces: is the dark stuff just shadow?". Icarus . 289: 281–294. doi:10.1016/j.icarus.2016.10.018. PMC 6839776 . PMID 31708591.

[3] See for example P.G.J Irwin; et al. (Jan 12, 2022). "Hazy Blue Worlds: A Holistic Aerosol Model for Uranus and Neptune, Including Dark Spots". Journal of Geophysical Research: Planets. 127 (6): e2022JE007189. arXiv: 2201.04516 . Bibcode: 2022JGRE..12707189I . doi: 10.1029/2022JE007189 . hdl: 1983/65ee78f0-1d28-4017-bbd9-1b49b24700d7 . PMC 9286428 . PMID 35865671. S2CID 245877540 .

[4] E. Hecht (2001). Optics (4th ed.). Pearson Education. ISBN 0-8053-8566-5.

[GoldBook-5] IUPAC , Compendium of Chemical Terminology , 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) " Reflectance ". doi : 10.1351/goldbook.R05235

[ILV-6] "CIE International Lighting Vocabulary". Archived from the original on 2016-06-16. Retrieved 2010-12-04.

[Grant-7] Palmer and Grant, The Art of Radiometry

[Ottav-8] Ottaviani, M. and Stamnes, K. and Koskulics, J. and Eide, H. and Long, S.R. and Su, W. and Wiscombe, W., 2008: 'Light Reflection from Water Waves: Suitable Setup for a Polarimetric Investigation under Controlled Laboratory Conditions . Journal of Atmospheric and Oceanic Technology, 25 (5), 715--728.

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Reflectance

Contents

Mathematical definitions

Hemispherical reflectance

Spectral hemispherical reflectance

Directional reflectance

Spectral directional reflectance

Reflectivity

Surface type

Water reflectance

Grating efficiency

Other radiometric coefficients

See also

Related Research Articles

References

External links